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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfsumf1o 14301* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)
(𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑛𝐶 𝐷)

Theoremsumss 14302* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑𝐵 ⊆ (ℤ𝑀))       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremfsumss 14303* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Theoremsumss2 14304* Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)
(((𝐴𝐵 ∧ ∀𝑘𝐴 𝐶 ∈ ℂ) ∧ (𝐵 ⊆ (ℤ𝑀) ∨ 𝐵 ∈ Fin)) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 if(𝑘𝐴, 𝐶, 0))

Theoremfsumcvg2 14305* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumsers 14306* Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 21-Apr-2014.)
((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → Σ𝑘𝐴 𝐵 = (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumcvg3 14307* A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 0))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Theoremfsumser 14308* A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 14320 and fsump1i 14342, which should make our notation clear and from which, along with closure fsumcl 14311, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = 𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁))

Theoremfsumcl2lem 14309* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremfsumcllem 14310* - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 3-Jun-2014.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 0 ∈ 𝑆)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremfsumcl 14311* Closure of a finite sum of complex numbers 𝐴(𝑘). (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℂ)

Theoremfsumrecl 14312* Closure of a finite sum of reals. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ)

Theoremfsumzcl 14313* Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℤ)

Theoremfsumnn0cl 14314* Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ0)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℕ0)

Theoremfsumrpcl 14315* Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → Σ𝑘𝐴 𝐵 ∈ ℝ+)

Theoremfsumzcl2 14316* A finite sum with integer summands is an integer. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘𝐴 𝐵 ∈ ℤ) → Σ𝑘𝐴 𝐵 ∈ ℤ)

Theoremfsumadd 14317* The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))

Theoremfsumsplit 14318* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 22-Apr-2014.)
(𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝑈 𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremsumsn 14319* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremfsum1 14320* The finite sum of 𝐴(𝑘) from 𝑘 = 𝑀 to 𝑀 (i.e. a sum with only one term) is 𝐵 i.e. 𝐴(𝑀). (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵)

Theoremsumpr 14321* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(𝑘 = 𝐴𝐶 = 𝐷)    &   (𝑘 = 𝐵𝐶 = 𝐸)    &   (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   (𝜑 → (𝐴𝑉𝐵𝑊))    &   (𝜑𝐴𝐵)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))

Theoremsumtp 14322* A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)
(𝑘 = 𝐴𝐷 = 𝐸)    &   (𝑘 = 𝐵𝐷 = 𝐹)    &   (𝑘 = 𝐶𝐷 = 𝐺)    &   (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))    &   (𝜑 → (𝐴𝑉𝐵𝑊𝐶𝑋))    &   (𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺))

Theoremsumsns 14323* A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.)
((𝑀𝑉𝑀 / 𝑘𝐴 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)

Theoremfsumm1 14324* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Theoremfzosump1 14325* Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀..^𝑁)𝐴 + 𝐵))

Theoremfsum1p 14326* Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑀𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))

Theoremfsummsnunz 14327* A finite sum with an additional integer summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ)

Theoremfsumsplitsnun 14328* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ 𝑧𝐴 ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 = (Σ𝑘𝐴 𝐵 + 𝑧 / 𝑘𝐵))

Theoremfsump1 14329* The addition of the next term in a finite sum of 𝐴(𝑘) is the current term plus 𝐵 i.e. 𝐴(𝑁 + 1). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝑁)𝐴 + 𝐵))

Theoremisumclim 14330* An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵)       (𝜑 → Σ𝑘𝑍 𝐴 = 𝐵)

Theoremisumclim2 14331* A converging series converges to its infinite sum. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘𝑍 𝐴)

Theoremisumclim3 14332* The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = Σ𝑘 ∈ (𝑀...𝑗)𝐴)       (𝜑𝐹 ⇝ Σ𝑘𝑍 𝐴)

Theoremsumnul 14333* The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 = ∅)

Theoremisumcl 14334* The sum of a converging infinite series is a complex number. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℂ)

Theoremisummulc2 14335* An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐵 · Σ𝑘𝑍 𝐴) = Σ𝑘𝑍 (𝐵 · 𝐴))

Theoremisummulc1 14336* An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (Σ𝑘𝑍 𝐴 · 𝐵) = Σ𝑘𝑍 (𝐴 · 𝐵))

Theoremisumdivc 14337* An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (Σ𝑘𝑍 𝐴 / 𝐵) = Σ𝑘𝑍 (𝐴 / 𝐵))

Theoremisumrecl 14338* The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 𝐴 ∈ ℝ)

Theoremisumge0 14339* An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ((𝜑𝑘𝑍) → 0 ≤ 𝐴)       (𝜑 → 0 ≤ Σ𝑘𝑍 𝐴)

Theoremisumadd 14340* Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘𝑍 (𝐴 + 𝐵) = (Σ𝑘𝑍 𝐴 + Σ𝑘𝑍 𝐵))

Theoremsumsplit 14341* Split a sum into two parts. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (𝐴𝐵) ⊆ 𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐶, 0))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = if(𝑘𝐵, 𝐶, 0))    &   ((𝜑𝑘 ∈ (𝐴𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )       (𝜑 → Σ𝑘 ∈ (𝐴𝐵)𝐶 = (Σ𝑘𝐴 𝐶 + Σ𝑘𝐵 𝐶))

Theoremfsump1i 14342* Optimized version of fsump1 14329 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝑍 = (ℤ𝑀)    &   𝑁 = (𝐾 + 1)    &   (𝑘 = 𝑁𝐴 = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝐾𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆))    &   (𝜑 → (𝑆 + 𝐵) = 𝑇)       (𝜑 → (𝑁𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇))

Theoremfsum2dlem 14343* Lemma for fsum2d 14344- induction step. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → ¬ 𝑦𝑥)    &   (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   (𝜓 ↔ Σ𝑗𝑥 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)       ((𝜑𝜓) → Σ𝑗 ∈ (𝑥 ∪ {𝑦})Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)

Theoremfsum2d 14344* Write a double sum as a sum over a two-dimensional region. Note that 𝐵(𝑗) is a function of 𝑗. (Contributed by Mario Carneiro, 27-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)

Theoremfsumxp 14345* Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑧 ∈ (𝐴 × 𝐵)𝐷)

Theoremfsumcnv 14346* Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)    &   (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → Rel 𝐴)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑥𝐴 𝐵 = Σ𝑦 𝐴𝐶)

Theoremfsumcom2 14347* Interchange order of summation. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)

Theoremfsumcom2OLD 14348* Obsolete proof of fsumcom2 14347 as of 2-Aug-2021. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐸 = Σ𝑘𝐶 Σ𝑗𝐷 𝐸)

Theoremfsumcom 14349* Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 𝐶 = Σ𝑘𝐵 Σ𝑗𝐴 𝐶)

Theoremfsum0diaglem 14350* Lemma for fsum0diag 14351. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁𝑘))))

Theoremfsum0diag 14351* Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 𝑀𝑗, 𝑀𝑘, 𝑗 + 𝑘𝑁." (Contributed by NM, 31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗)))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...(𝑁𝑘))𝐴)

Theoremmptfzshft 14352* 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. Formerly part of proof for fsumshft 14354. (Contributed by AV, 24-Aug-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝑗 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)) ↦ (𝑗𝐾)):((𝑀 + 𝐾)...(𝑁 + 𝐾))–1-1-onto→(𝑀...𝑁))

Theoremfsumrev 14353* Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)

Theoremfsumshft 14354* Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV, 8-Sep-2019.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝑘𝐾) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)

Theoremfsumshftm 14355* Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝑘 + 𝐾) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝑀𝐾)...(𝑁𝐾))𝐵)

Theoremfsumrev2 14356* Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = ((𝑀 + 𝑁) − 𝑘) → 𝐴 = 𝐵)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...𝑁)𝐵)

Theoremfsum0diag2 14357* Two ways to express "the sum of 𝐴(𝑗, 𝑘) over the triangular region 0 ≤ 𝑗, 0 ≤ 𝑘, 𝑗 + 𝑘𝑁." (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝑥 = 𝑘𝐵 = 𝐴)    &   (𝑥 = (𝑘𝑗) → 𝐵 = 𝐶)    &   ((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗)))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (0...𝑁𝑘 ∈ (0...(𝑁𝑗))𝐴 = Σ𝑘 ∈ (0...𝑁𝑗 ∈ (0...𝑘)𝐶)

Theoremfsummulc2 14358* A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (𝐶 · Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐶 · 𝐵))

Theoremfsummulc1 14359* A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (Σ𝑘𝐴 𝐵 · 𝐶) = Σ𝑘𝐴 (𝐵 · 𝐶))

Theoremfsumdivc 14360* A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (Σ𝑘𝐴 𝐵 / 𝐶) = Σ𝑘𝐴 (𝐵 / 𝐶))

Theoremfsumneg 14361* Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 -𝐵 = -Σ𝑘𝐴 𝐵)

Theoremfsumsub 14362* Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 (𝐵𝐶) = (Σ𝑘𝐴 𝐵 − Σ𝑘𝐴 𝐶))

Theoremfsum2mul 14363* Separate the nested sum of the product 𝐶(𝑗) · 𝐷(𝑘). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐵) → 𝐷 ∈ ℂ)       (𝜑 → Σ𝑗𝐴 Σ𝑘𝐵 (𝐶 · 𝐷) = (Σ𝑗𝐴 𝐶 · Σ𝑘𝐵 𝐷))

Theoremfsumconst 14364* The sum of constant terms (𝑘 is not free in 𝐴). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑘𝐴 𝐵 = ((#‘𝐴) · 𝐵))

Theoremmodfsummodslem1 14365* Lemma 1 for modfsummods 14366. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → 𝑧 / 𝑘𝐵 ∈ ℤ)

Theoremmodfsummods 14366* Induction step for modfsummod 14367. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ((Σ𝑘𝐴 𝐵 mod 𝑁) = (Σ𝑘𝐴 (𝐵 mod 𝑁) mod 𝑁) → (Σ𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 mod 𝑁) = (Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) mod 𝑁)))

Theoremmodfsummod 14367* A finite sum modulo a positive integer equals the finite sum of their summands modulo the positive integer, modulo the positive integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℤ)       (𝜑 → (Σ𝑘𝐴 𝐵 mod 𝑁) = (Σ𝑘𝐴 (𝐵 mod 𝑁) mod 𝑁))

Theoremfsumge0 14368* If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)       (𝜑 → 0 ≤ Σ𝑘𝐴 𝐵)

Theoremfsumless 14369* A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by NM, 26-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → Σ𝑘𝐶 𝐵 ≤ Σ𝑘𝐴 𝐵)

Theoremfsumge1 14370* A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   (𝑘 = 𝑀𝐵 = 𝐶)    &   (𝜑𝑀𝐴)       (𝜑𝐶 ≤ Σ𝑘𝐴 𝐵)

Theoremfsum00 14371* A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)       (𝜑 → (Σ𝑘𝐴 𝐵 = 0 ↔ ∀𝑘𝐴 𝐵 = 0))

Theoremfsumle 14372* If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 ≤ Σ𝑘𝐴 𝐶)

Theoremfsumlt 14373* If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐵 < 𝐶)       (𝜑 → Σ𝑘𝐴 𝐵 < Σ𝑘𝐴 𝐶)

Theoremfsumabs 14374* Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))

Theoremtelfsumo 14375* Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = 𝑁𝐴 = 𝐸)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵𝐶) = (𝐷𝐸))

Theoremtelfsumo2 14376* Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = 𝑁𝐴 = 𝐸)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶𝐵) = (𝐸𝐷))

Theoremtelfsum 14377* Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → (𝑁 + 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐵𝐶) = (𝐷𝐸))

Theoremtelfsum2 14378* Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
(𝑘 = 𝑗𝐴 = 𝐵)    &   (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶)    &   (𝑘 = 𝑀𝐴 = 𝐷)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → (𝑁 + 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)(𝐶𝐵) = (𝐸𝐷))

Theoremfsumparts 14379* Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
(𝑘 = 𝑗 → (𝐴 = 𝐵𝑉 = 𝑊))    &   (𝑘 = (𝑗 + 1) → (𝐴 = 𝐶𝑉 = 𝑋))    &   (𝑘 = 𝑀 → (𝐴 = 𝐷𝑉 = 𝑌))    &   (𝑘 = 𝑁 → (𝐴 = 𝐸𝑉 = 𝑍))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝑉 ∈ ℂ)       (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐵 · (𝑋𝑊)) = (((𝐸 · 𝑍) − (𝐷 · 𝑌)) − Σ𝑗 ∈ (𝑀..^𝑁)((𝐶𝐵) · 𝑋)))

Theoremfsumrelem 14380* Lemma for fsumre 14381, fsumim 14382, and fsumcj 14383. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐹:ℂ⟶ℂ    &   ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))       (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))

Theoremfsumre 14381* The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (ℜ‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (ℜ‘𝐵))

Theoremfsumim 14382* The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (ℑ‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (ℑ‘𝐵))

Theoremfsumcj 14383* The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (∗‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (∗‘𝐵))

Theoremfsumrlim 14384* Limit of a finite sum of converging sequences. Note that 𝐶(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐷(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)    &   ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ⇝𝑟 𝐷)       (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)

Theoremfsumo1 14385* The finite sum of eventually bounded functions (where the index set 𝐵 does not depend on 𝑥) is eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 22-May-2016.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)    &   ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝑂(1))       (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))

Theoremo1fsum 14386* If 𝐴(𝑘) is O(1), then Σ𝑘𝑥, 𝐴(𝑘) is O(𝑥). (Contributed by Mario Carneiro, 23-May-2016.)
((𝜑𝑘 ∈ ℕ) → 𝐴𝑉)    &   (𝜑 → (𝑘 ∈ ℕ ↦ 𝐴) ∈ 𝑂(1))       (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))𝐴 / 𝑥)) ∈ 𝑂(1))

Theoremseqabs 14387* Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by Mario Carneiro, 26-Mar-2014.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑 → (abs‘(seq𝑀( + , 𝐹)‘𝑁)) ≤ (seq𝑀( + , 𝐺)‘𝑁))

Theoremiserabs 14388* Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) = (abs‘(𝐹𝑘)))       (𝜑 → (abs‘𝐴) ≤ 𝐵)

Theoremcvgcmp 14389* A comparison test for convergence of a real infinite series. Exercise 3 of [Gleason] p. 182. (Contributed by NM, 1-May-2005.) (Revised by Mario Carneiro, 24-Mar-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐺𝑘))    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ≤ (𝐹𝑘))       (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Theoremcvgcmpub 14390* An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)    &   (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ≤ (𝐹𝑘))       (𝜑𝐵𝐴)

Theoremcvgcmpce 14391* A comparison test for convergence of a complex infinite series. (Contributed by NM, 25-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐺𝑘)) ≤ (𝐶 · (𝐹𝑘)))       (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Theoremabscvgcvg 14392* An absolutely convergent series is convergent. (Contributed by Mario Carneiro, 28-Apr-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (abs‘(𝐺𝑘)))    &   ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Theoremclimfsum 14393* Limit of a finite sum of converging sequences. Note that 𝐹(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐵(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Mario Carneiro, 22-May-2016.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐹𝐵)    &   (𝜑𝐻𝑊)    &   ((𝜑 ∧ (𝑘𝐴𝑛𝑍)) → (𝐹𝑛) ∈ ℂ)    &   ((𝜑𝑛𝑍) → (𝐻𝑛) = Σ𝑘𝐴 (𝐹𝑛))       (𝜑𝐻 ⇝ Σ𝑘𝐴 𝐵)

Theoremfsumiun 14394* Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘 𝑥𝐴 𝐵𝐶 = Σ𝑥𝐴 Σ𝑘𝐵 𝐶)

Theoremhashiun 14395* The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ Fin)    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → (#‘ 𝑥𝐴 𝐵) = Σ𝑥𝐴 (#‘𝐵))

Theoremhashrabrex 14396* The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
(𝜑𝑌 ∈ Fin)    &   ((𝜑𝑦𝑌) → {𝑥𝑋𝜓} ∈ Fin)    &   (𝜑Disj 𝑦𝑌 {𝑥𝑋𝜓})       (𝜑 → (#‘{𝑥𝑋 ∣ ∃𝑦𝑌 𝜓}) = Σ𝑦𝑌 (#‘{𝑥𝑋𝜓}))

Theoremhashuni 14397* The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ Fin)    &   (𝜑Disj 𝑥𝐴 𝑥)       (𝜑 → (#‘ 𝐴) = Σ𝑥𝐴 (#‘𝑥))

Theoremqshash 14398* The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝜑 Er 𝐴)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → (#‘𝐴) = Σ𝑥 ∈ (𝐴 / )(#‘𝑥))

Theoremackbijnn 14399* Translate the Ackermann bijection ackbij1 8943 onto the positive integers. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝐹 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑦𝑥 (2↑𝑦))       𝐹:(𝒫 ℕ0 ∩ Fin)–1-1-onto→ℕ0

5.10.4  The binomial theorem

Theorembinomlem 14400* Lemma for binom 14401 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁𝑘)) · (𝐵𝑘))))       ((𝜑𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵𝑘))))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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